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Table of Contents

Section Page

Assembly

ME-8951 Rotating Platform ...................................................................... 3

ME-8953 Rotational Inertia Accessory ..................................................... 4

Experiments Using the ME-8951 Rotating Platform

Exp 1 Rotational Inertia of a Point Mass....................................................7

Experiments Using the ME-8952 Centripetal Force Accesory

Exp 2 Conservation of Angular Momentum Using Point Mass ............... 11

Experiments Using the ME-8953 Rotational Inertia Accessory

Exp 3 Rotational Inertia of Disk and Ring ................................................ 16

Exp 4 Rotational Inertia of Disk Off-Axis (Fixed/Rotating)......................21

Exp 5 Conservation of Angular Momentum Using Disk and Ring... .........25

2

ME-8951 Rotating Platform Equipment

The ME-8951 Rotating Platform Includes the

following:

-PASCO cast iron “A” base with rotating shaft

and pulley with 10 holes

-aluminum track

-two square masses (about 300 g) with thumb

screw and square nut

-two additional low-profile screws and square nuts

to act as stops for the square mass in the Conser-

vation of Angular Momentum experiment

-accessory mounting rod for mounting the 10spoke

pulley or the optional Smart Pulley photo-gate head

-accessory mounting rod for mounting PASCO

Photogate (ME-9498A, ME-9402B or later)

Equipment

3

mass ring

ME-8953 Rotational Inertia Accessory Equipment

The ME-8953 Rotation Inertia Accessory includes:

-disk with bearings in the center

-ring (12.7 cm diameter)

-adapter to connect disk to platform

-10-spoke pulley and rod

Other Equipment Needed:

The following is a list of equipment recommended for

the experiments described in this manual. See the

PASCO catalog for more information.

-Projectile Launcher

-Projectile Collision Accessory

-Smart Pulley (with Smart Pulley Timer soft

ware, or a compatible computer interface)

-string -mass and hanger set

-balance (for measuring mass)

-calipers

-stopwatch

Miscellaneous Supplies:

-meter stick

-graph paper

-carbon paper

-white paper

-rubber bands

-paper clips

Assembly

4

rotating platform

rotating platform

Figure 2: Leveling the Base Leveling the Base

Some experiments (such as the Centripetal Force

experiments) require the apparatus to be extremely

level. If the track is not level, the uneven performance

will affect the results. To level the base, perform the

following steps:

1. Purposely make the apparatus unbalanced by at-

taching the 300 g square mass onto either end of

the aluminum track. Tighten the screw so the mass

will not slide. If the hooked mass is hanging from

the side post in the centripetal force accessory,

place the square mass on the same side.

2. Adjust the leveling screw on one of the legs of

the base until the end of the track with the square

mass is aligned over the leveling screw on the

other leg of the base. See Figure 2.

3. Rotate the track 90 degrees so it is parallel to one

side of the “A” and adjust the other leveling

screw until the track will stay in this position. 4. The track is now level and it should remain at

rest regardless of its orientation.

Installing the Optional Smart Pulley Photogate Head

The black plastic rod stand is designed to be used in

two ways:

It can be used to mount a Smart Pulley photogate

head to the base in the correct position to use the 10

holes in the pulley on the rotating shaft to measure

angular speed.

It can be used to mount a Smart Pulley (with the pul-

ley and rod) to the base to run a string over the pulley.

in the pulley. If the photogate head is powered by a

computer, you can tell when the photogate is blocked

by watching the LED indicator on the end of the

photogate. The photogate head should not be rubbing

against the pulley. When the head is in the correct

position, tighten the bottom screw to fix the rod in

place.

1. To install, first mount the black rod to the base by

(optional) inserting the rod into either hole adjacent to

the center shaft on the base.

2. Mount the Smart Pulley photogate head horizon-

tally with the cord side down. Align the screw hole

in the photogate head with the screw hole in the flat

side of the black rod. Secure the photogate head

with the thumb screw. See Figure 3.

3. Loosen the thumb screw on the base to allow the

black rod to rotate. Orient the rod and photogate

head so the infrared beam passes through the holes

To Use the Photogate Head Only:

Figure 3: Using the Accessory Mounting Rod With the Smart Pulley

300g square

mass

leveling

feet

Adjust this foot

first

300g square

mass Adjust this

foot Leveling feet

5

Figure 5: Rotational Inertia Accessory Including Platform Adapter Assembly

ME-8953 Rotational Inertia Accessory Rotational Inertia Accessory Assembly Little

assembly is required to use the Rotational Inertia

Accessory. The rotational disk can be placed directly

onto the axle of the rotating base or can be used with

the rotating platform via the included platform adapter.

Platform Adapter Assembly

1. Attach the square nut (supplied with the

Rotational Inertia Accessory) to the platform

adapter.

2. Position the platform adapter at the desired radius

as shown in Figure 5. 3. Grip the knurled edge of the platform adapter and

tighten.

The rotating disk can be mounted in a variety of positions

using any of the four holes on the rotation disk.

Two “D” holes exist on the edge of the disk, located at

180˚ from one another.

One “D” hole is located at the center on the top surface

(the surface with the metal ring channel and the

PASCO label) of the disk.

One hole is located at the center on the bottom surface

of the disk and is actually the inner race of a bearing.

This enables the rotational disk to rotate (in either

direction) in addition to other rotating motions applied

to your experiment setup.

Platform

adapter Rotating

Platform

To use the Super Pulley and Photogate Head

with the Pulley Mounting Rod:

1. Attach the Super Pulley -- and the Photogate

Head if needed -- to the Pulley Mounting Rod.

2. Insert the pulley mounting rod into the hole in

the Accessory Mounting Rod and tighten the

thumb screw. See Figure 4.

3. Rotate the accessory mounting rod so that a

string from the pulley on the center shaft will

be aligned with the groove on the Super

Pulley.

4. Adjust the position of the base so the string

passing over the Super Pulley will clear the

edge of the table. Figure 4: Using the Accessory Mounting Rod With the Pulley Mounting Rod, Super Pulley, and Photogate Head

platform

adapter rotating

platform

6

Theory

I

The purpose of this experiment is to find the rotational inertia of a point mass experimentally

and to verify that this value corresponds to the calculated theoretical value.

Theoretically, the rotational inertia, I, of a point mass is given by I = MR2, where M is the

mass, R is the distance the mass is from the axis of rotation.

To find the rotational inertia experimentally, a known torque is applied to the object and

the resulting angular acceleration is measured. Since τ = Iα,

where α is the angular acceleration which is equal to a/r and τ is the torque caused by the

weight hanging from the thread which is wrapped around the base of the apparatus.

τ = rT

where r is the radius of the cylinder about which the thread is wound and T is the tension

in the thread when the apparatus is rotating.

Applying Newton‟s Second Law for the hanging mass, m, gives (see Figure 1.1).

Σ F = mg – T = ma

Figure 1.1: Rotational Apparatus and Free-Body Diagram

Solving for the tension in the thread gives:

T = m (g-a)

EQUIPMENT REQUIRED

- Precision Timer Program - mass and hanger set

- paper clips (for masses < 1 g) - 10-spoke pulley with photogate head

- triple beam balance - calipers

Purpose

Experiment 1: Rotational Inertia of a Point Mass

7

Once the linear acceleration of the mass (m) is determined, the torque and the angular acceleration

can be obtained for the calculation of the rotational inertia.

Rota

platf

Setup

1. Attach the square mass (point mass) to the track on the rotating platform at any radius you wish.

2. Mount the Smart Pulley to the base and connect it to a computer. See Figure 1.2.

3. Run the Smart Pulley Timer program.

"A" base

Procedure

Part I: Measurements For the Theoretical Rotational Inertia

Table 1.1: Theoretical Rotational Inertia 1. Weigh the square mass to find the mass M

and record in Table 1.1.

2. Measure the distance from the axis of

rotation to the center of the square mass

and record this radius in Table 1.1.

Part II: Measurement For the Experimental Method

Accounting For Friction

Because the theory used to find the rotational inertia experimentally does not include friction, it will

be compensated for in this experiment by finding out how much mass over the pulley it takes to overcome

kinetic friction and allow the mass to drop at a constant speed. Then this “friction mass” will be subtracted

from the mass used to accelerate the ring.

1. Start the DataStudio program. Select ‟Smart Pulley (Linear)‟ and set up a Digits display to show

velocity with three significant figures.

2. Hang a small amount of mass - such as a few paper clips - on the end of the thread that is over the

pulley.

3. Start monitoring data, and then give the Rotating Platform a tap to get it started moving.

4. Watch the Digits display to see the velocity.

5. If the velocity increases or decreases as the platform turns, stop monitoring data, stop the platform,

and adjust the amount of mass on the thread by adding or removing a paper clip.

6. Repeat the process until the velocity stays constant as the mass falls.

7. Measure the mass on the end of the thread and record it as the ‟Friction Mass‟ in Table 1.2

Rotating

platform

10-spoke pulley with

photogate head

Figure 1.2: Rotational inertia of a point mass

Figure 1.2: Rotational inertia of a point mass

300g mass

8

Finding the Acceleration of the Point Mass and Apparatus

To find the acceleration, put about 50 g - measure the exact mass and record it in Table 1.2 - on the

end of the thread over the pulley. In DataStudio, set up a Graph display of Velocity versus Time.

1. Wind the thread up and hold the Rotating Platform.

2. Let the platform begin to turn and at the same time, start recording data.

3. Let the mass fall toward the floor but STOP recording data just before the mass hits the floor.

4. Examine your Graph display of Velocity versus Time. The slope of the best ‟Linear Fit‟ for your data

is the acceleration of the apparatus.

5. Record the slope in Table 1.2

.Table 1.2: Rotational Inertia Data

Measure the Radius

1. Using calipers, measure the diameter of the cylinder about which the thread is wrapped and

calculate the radius. Record in Table 1.2.

Finding the Acceleration of the Apparatus Alone

Since in Finding the Acceleration of the Point Mass and Apparatus the apparatus is rotating as

well as the point mass, it is necessary to determine the acceleration, and the rotational inertia, of

the apparatus by itself so this rotational inertia can be subtracted from the total, leaving only the

rotational inertia of the point mass.

1. Take the point mass off the rotational apparatus and repeat Finding the Acceleration of the

Point Mass and Apparatus for the apparatus alone.

➤ NOTE: that it will take less “friction mass” to overcome the new kinetic friction and it is only

necessary to put about 20 g over the pulley in Finding the Acceleration of the Point Mass and

Apparatus.

2. Record the data in Table 1.2.

Calculations

1. Subtract the “friction mass” from the hanging mass used to accelerate the apparatus to determine

the mass, m, to be used in the equations.

2. Calculate the experimental value of the rotational inertia of the point mass and apparatus together


3. Calculate the experimental value of the rotational inertia of the apparatus alone. Record in Table

1.3.

Point Mass and Apparatus Apparatus Alone

Friction Mass

Hanging Mass

Slope

Radius

9

4. Subtract the rotational inertia of the apparatus from the combined rotational inertia of the point

mass and apparatus. This will be the rotational inertia of the point mass alone. Record in Table

5. Calculate the theoretical value of the rotational inertia of the point mass. Record in Table 1.3.

6. Use a percent difference to compare the experimental value to the theoretical value. Record in

Table 1.3

Table 1.3: Results

Rotational Inertia for Point Mass and Apparatus Combined

Rotational Inertia for Apparatus Alone

Rotational Inertia for Point Mass (experimental value)

Rotational Inertia for Point Mass (theoretical value)

% Difference

10

EQUIPMENT REQUIRED

-Smart Pulley Timer Program

-Rotational Inertia Accessory (ME-8953)

-Rotating Platform (ME-8951)

-Smart Pulley

-balance

Purpose

A mass rotating in a circle is pulled in to a smaller radius and the new angular speed is

predicted using conservation of angular momentum.

Theory

Angular momentum is conserved when the radius of the circle is changed.

L = Iiω i = If ω f

where Ii is the initial rotational inertia and ω

i is the initial angular speed. So the final rotational

speed is given by:

Ii ω

f = ω

i If

To find the rotational inertia experimentally, a known torque is applied to the object and the

resulting angular acceleration is measured. Since τ = Iα,

I

where α is the angular acceleration which is equal to a/r and τ is the torque caused by the weight

hanging from the thread which is wrapped around the base of the apparatus.

τ = rT

where r is the radius of the cylinder about which the thread is wound and T is the tension in the

thread when the apparatus is rotating.

Applying Newton‟s Second Law for the hanging mass, m, gives (See Figure 2.1)



T = m(g-a)



Experiment 2: Conservation of Angular Momentum Using Point Mass

EQUIPMENT REQUIRED

11


Part I: Conservation of Angular Momentum

Setup

1. Level the apparatus using the square mass on the track as shown in the leveling instructions in the

Assembly Section.

2. Slide a thumb screw and square nut into the T-slot on the top of the track and tighten it down at

about the 5 cm mark. This will act as a stop for the sliding square mass. See Figure 2.2.

Figure 2.2: Set-up for conservation of angular momentum

3. With the side of the square mass that has the hole oriented toward the center post, slide the square

mass onto the track by inserting its square nut into the T-slot, but do not tighten the thumb screw;

the square mass should be free to slide in the T-slot.

4. Slide a second thumb screw and square nut into the T-slot and tighten it down at about the 20 cm

mark. Now the square mass is free to slide between the two limiting stops.

5. Move the pulley on the center post to its lower position. Remove the spring bracket from the center

post and set it aside.

6. Attach a string to the hole in the square mass and thread it around the pulley on the center post and

pass it through the indicator bracket.

7. Mount the Smart Pulley photogate on the black rod on the base and position it so it straddles the

holes in the pulley on the center rotating shaft.


12

Procedure

1. Select ‟Smart Pulley (Rotational)‟ as the type of sensor. Set up a Graph display of Velocity (rad/s) versus

time.

2. Hold the string just above the center post. With the square mass against the outer stop, give the track a spin

using your hand.

3. Click ‟Start‟ to begin recording data. After about 20 data points have been taken, pull up on the string to

cause the square mass to slide from the outer stop to the inner stop.

4. Continue to hold the string up and take about 20 data points after pulling up on the string. Click ‟Stop‟ to

end recording data.

5. Examine the Graph display of Velocity (rad/s) versus

time. The graph shows the angular speed before and after

the square mass is pulled toward the inner stop. Rescale

the graph if necessary.

6. Use the Smart Cursor tool to determine the angular

speed immediately before and immediately after pulling

the string. Record these values in Table 2.1.

7. Repeat the experiment a total of three times with

different initial angular speeds. Record these values

in Table 2.1

Table 2.1: Data

Part II: Determining the Rotational Inertia

Measure the rotational inertia of the apparatus twice: once with the square mass in its initial

position and once with it in its final position

Setup

1. Attach a Smart Pulley with rod to the base using the black rod. 2. Wind a thread around the pulley on the center shaft and pass the thread over the Smart

Pulley. See Figure 2.3.


photogate head

hanging mass

Angular Speeds

Trial Number Initial Final

1

2

3

Figure 2.3: Set-up for determining of rotational inertia

13

Procedure


Because the theory used to find the rotational inertia experimentally does not include friction, it will be

compensated for in this experiment by finding out how much mass over the pulley it takes to overcome


from the mass used to accelerate the apparatus.

1. Start the DataStudio program. Select ‟Smart Pulley (Linear)‟ and set up a Digits display to how velocity

with three significant figures.

2. Hang a small amount of mass (such as a few paper clips) on the end of the thread that is over the pulley.

Make sure that the thread is wound around the step pulley.

3. Start monitoring data, and then give the Rotating Platform a tap to get it started moving.


5. If the velocity increases or decreases as the platform turns, stop monitoring data, stop the platform, and

adjust the amount of mass on the end of the thread.

6. Repeat the process until the velocity stays constant.

7. Measure the mass on the end of the thread and record it as ‟Friction Mass‟ in Table 2.2..

Finding the Acceleration of the Apparatus

To find the acceleration, put about 30 g - record the exact hanging mass in Table 2.2 - over the pulley. In the

DataStudio program, set up a Graph display of Velocity versus Time.


2. Let the Rotating Platform begin to turn and at the same time, START recording data.

3. Let the mass descend toward the floor but STOP recording data just before the mass hits the floor.

4. Examine your graph of velocity versus time. The slope ("m") of the best fit line for your data is the

acceleration (use Fit>Linear Fit). Record the slope in Table 2.2. Repeat the procedure for the mass at the

inner stop. Record results in Table 2.2.

Measure Radius

1. Using calipers, measure the diameter of the step pulley about which the thread is wrapped and calculate

the radius.

2. Record the radius in Table 2.2.Table

2.2 Rotational Inertia Data

Mass at Outer Stop Mass at Inner Stop

Friction Mass

Hanging Mass

Slope

Radius

Rotational Inertia

14

Analysis

1. Calculate the rotational inertia‟s:

Subtract the “friction mass” from the hanging mass used to accelerate the apparatus to

determine the mass, m, to be used in the equations.

Calculate the experimental values of the rotational inertia and record it in Table 2.3.

2. Calculate the expected (theoretical) values for the final angular velocity and record these

values in Table 2.3.

Table 2.3: Results

2

2

1iii IKE Calculate the rotational Kinetic energy before the string was pulled. Calculate the rotational Kinetic energy

Then calculate the rotational Kinetic

energy

2

2

1iii IKE

2

2

1fff IKE

after the string was pulled.

Which kinetic energy is greater? Why?

3. For each trial, calculate the percent difference between the experimental and the

theoretical values of the final angular velocity and record these in Table 2.3.

Questions

Trial #1 Trial #2 Trial #3

Theoretical Angular Speed

% Difference

15

)(2

1 2

2

2

1 RRMI

2

2

1MRI

2

4

1MRI

I

The purpose of this experiment is to find the rotational inertia of a ring and a disk

experimentally and to verify that these values correspond to the calculated theoretical values.

Theory

Theoretically, the rotational inertia, I, of a ring about its center of mass is

given by: R1

where M is the mass of the ring, R1 is the inner radius of the ring, and R

2 is

the outer radius of the ring. See Figure 3.1.

The rotational inertia of a disk about its center of mass is given by:

where M is the mass of the disk and R is the radius of the disk. The rotational inertia of a

disk about its diameter is given by:

Disk about center of Mass Disk about Diameter

Figure 3.2:

To find the rotational inertia experimentally, a known torque is applied to the object and

the resulting angular acceleration is measured. Since τ = Iα,

EQUIPMENT REQUIRED


- Rotational Inertia Accessory (ME-9341) - paper clips (for masses < 1 g)

- Smart Pulley - triple beam balance

- calipers

Purpose

Experiment 3: Rotational Inertia of Disk and Ring

R1

Figure 3.1: Ring

16

where α is the angular acceleration which is equal to a/r and τ is the torque caused by the weight

hanging from the thread which is wrapped around the base of the apparatus.

τ = rT

where r is the radius of the cylinder about which the thread is wound and T is the tension in the

thread when the apparatus is rotating.



T

rotational

disk

a

hanging mg mass



T = m(g-a)



Setup

1. Remove the track from the Rotating Platform and

place the disk directly on the center shaft as shown in

Figure 3.4. The side of the disk that has the

indentation for the ring should be up.

2. Place the ring on the disk, seating it in this

indentation.

3. Mount the Photogate/Pulley System to the base and

connect it to a PASCO interface.

4. Attach a thread to the top step of the three-step pulley

on the Rotational Apparatus shaft and suspend the

string over the pulley of the Photogate/Pulley

System. Attach a hanger and mass to the end of

the thread.

5. Start the DataStudio program.

hanger

Procedure

Measurements for the Theoretical Rotational Inertia

1. Weigh the ring and disk to find their masses and record these masses in Table 3.1.

2. Measure the inside and outside diameters of the ring and calculate the radii R1 and R2. Record in Table 3.1.

3. Measure the diameter of the disk and calculate the radius R and record it in Table 3.1.


photogate head

mass ring mass and

hanger

Figure 3.4: Set-up for Disk and Ring

17

Table 3.1: Theoretical Rotational Inertia

Measurements for the Experimental Method






1. In the DataStudio program, select ‟Smart Pulley (Linear)‟ and set up a Digits display to show velocity


2. Hang a small amount of mass such as a few paper clips on the end of the thread that is over the pulley.

3. Start monitoring data, and then give the Rotational Disk a tap to get it started moving.


5. If the velocity increases or decreases as the Rotational Disk turns, stop monitoring data, stop the

Rotational Disk, and adjust the amount of mass on the thread by adding or removing a paper clip.


7. Measure the mass on the end of the thread and record it as the ‟Friction Mass‟ in Table 3.2

Table 3.2: Rotational Inertia Data

Finding the Acceleration of Ring and Disk

To find the acceleration, put about 50 g - record the exact hanging mass in Table 3.2 - over the pulley.

In the DataStudio program, set up a Graph display of Velocity versus Time.


2. Let the Rotating Platform begin to turn and at the same time, start recording data.


4. Examine your Graph display of Velocity versus Time. The slope of the best fit line for your data is the

acceleration of the apparatus.

5. Record the slope in Table 3.2.

Mass of Ring

Mass of Disk

Inner Radius of Ring

Outer Radius of Ring

Radius of Disk

Ring and Disk Combined

Disk Alone Disk Vertical

Friction Mass

Hanging Mass

Slope

Radius

18

Measure the Radius

1. Using calipers, measure the diameter of the cylinder about which the thread is wrapped and calculate the

radius. Record in Table 3.2.

Finding the Acceleration of the Disk Alone

Since in Finding the Acceleration of Ring and Disk the disk is rotating as well as the ring, it is

necessary to determine the acceleration, and the rotational inertia, of the disk by itself so this

rotational inertia can be subtracted from the total, leaving only the rotational inertia of the ring.

1. To do this, take the ring off the rotational apparatus and repeat Finding the Acceleration of Ring

and Disk for the disk alone.

➤ NOTE: that it will take less “friction mass” to overcome the new kinetic friction and it is only

necessary to put about 30 g over the pulley in Finding the Acceleration of Ring and Disk.

Disk Rotating on an Axis Through Its Diameter

Remove the disk from the shaft and rotate it up on its side. Mount the disk vertically by inserting the

shaft in one of the two “D”-shaped holes on the edge of the disk. See Figure 3.5.

➤ WARNING! Never mount the disk vertically using the adapter on the track. The adapter is too short

for this purpose and the disk might fall over while being rotated.

Repeat steps Measure the Radius and Finding the Acceleration of the Disk Alone to determine the

rotational inertia of the disk about its diameter. Record the data in Table 3.2.

Figure 3.5: Disk mounted vertically

Calculations

Record the results of the following calculations in Table 3.3.



2. Calculate the experimental value of the rotational inertia of the ring and disk together.

3. Calculate the experimental value of the rotational inertia of the disk alone.

4. Subtract the rotational inertia of the disk from the total rotational inertia of the ring and disk. This

will be the rotational inertia of the ring alone.

19

5. Calculate the experimental value of the rotational inertia of the disk about its diameter.

6. Calculate the theoretical value of the rotational inertia of the ring.

7. Calculate the theoretical value of the rotational inertia of the disk about its center of mass and

about its diameter

8. Use a percent difference to compare the experimental values to the theoretical values.

Table 3.3: Results

Rotational Inertia for Ring and Disk Combined

Rotational Inertia for Disk Alone (experimental value)

Rotational Inertia for Ring (experimental value)

Rotational Inertia for Vertical Disk (experimental value)

Rotational Inertia for Disk (theoretical value)

Rotational Inertia for Ring (theoretical value)

Rotational Inertia for Vertical Disk (theoretical value)

% Difference for Disk

% Difference for Ring

% Difference for Vertical Disk

20

Theory

Icm = MR2

I = Icm + Md2

I

The purpose of this experiment is to find the rotational inertia of a disk about an axis parallel to the

center of mass axis.

Theoretically, the rotational inertia, I, of a disk about a perpendicular axis through its center of

mass is given by:

where M is the mass of the disk and R is the radius of the disk. The rotational inertia of a disk

about an axis parallel to the center of mass axis is given by:

where d is the distance between the two axes.

In one part of this experiment, the disk is mounted on its ball bearing side which allows the disk

to freely rotate relative to the track. So as the track is rotated, the disk does not rotate relative to

its center of mass. Since the disk is not rotating about its center of mass, it acts as a point mass

rather than an extended object and its rotational inertia reduces from:

I = Icm+ Md2

to I = Md2

To find the rotational inertia experimentally, a known torque is applied to the object and the

resulting angular acceleration is measured. Since τ = Iα,

where α is the angular acceleration which is equal to a/r and τ is the torque caused by the

weight hanging from the thread which is wrapped around the base of the apparatus.

τ = rT

where r is the radius of the cylinder about which the thread is wound and T is the tension

in the thread when the apparatus is rotating.

EQUIPMENT REQUIRED


- Rotational Inertia Accessory (ME-8953) - paper clips (for masses < 1 g)

- Smart Pulley - triple beam balance

- calipers

Purpose

Experiment 4: Rotational inertia of Disk Off-Axis (Fixed/Rotating)

21



rotational disk

a


T = m(g-a)

Once the linear acceleration of the mass (m) is determined, the torque and the angular acceleration can

be obtained for the calculation of the rotational inertia.

Setup

1. Set up the Rotational Accessory as shown in Figure 4.2. Mount the disk with its bearing side up.

Use the platform adapter to fasten the disk to the track at a large radius.

2. Mount the Smart Pulley to the base and connect it to a computer.


4. Run the DataStudio program.

Measurements For the Theoretical Rotational Inertia

Record these measurements in Table 4.1.

1. Weigh the disk to find the mass M.

2. Measure the diameter and calculate the radius R

3. Measure the distance, d, from the axis of rotation to the center of the disk.

Figure 4.1: Rotational Apparatus and Free-body Diagram

rotating

disk

rotational

disk

Figure 4.2: Set-up for Dick Off-Axis


22

Table 4.1: Theoretical Rotational Inertia

Measurements for the Experimental Method




kinetic friction and allow the mass to drop at a constant speed. Then this “friction mass” will be

subtracted


1. In the DataStudio program, select ‟Smart Pulley (Linear)‟ and set up a Digits display to show velocity


2. Hang a small amount of mass such as a few paper clips on the end of the thread that is over the

pulley.

3. Start monitoring data, and then give the Rotational Disk a tap to get it started moving.


5. If the velocity increases or decreases as the Rotational Disk turns, stop monitoring data, stop the

Rotational Disk, and adjust the amount of mass on the thread by adding or removing a paper clip.


7. Measure the mass on the end of the thread and record it as the ‟Friction Mass‟ in Table 4.2.

Table 4.2: Rotational Inertia Data

Finding the Acceleration of Ring and Disk

To find the acceleration, put about 50 g - record the exact hanging mass in Table 3.2 - over the pulley.

In the DataStudio program, set up a Graph display of Velocity versus Time.


2. Let the Rotating Platform begin to turn and at the same time, start recording data.


4. Examine your Graph display of Velocity versus Time. The slope of the best fit line for your data is

the acceleration of the apparatus.

5. Record the slope in Table 4.2.

Mass of Disk

Radius of Disk

Distance Between Parallel Axis

Fixed Disk and Track Combined

Track Alone Rotating Disk and Track Combined

Friction Mass

Hanging Mass

Slope

Radius

23

Measure the Radius

1. Using calipers, measure the diameter of the cylinder about which the thread is wrapped and

calculate the radius. Record in Table 4.2.

Finding the Acceleration of Track Alone

Since in Finding the Acceleration of Disk and Track the track is rotating as well as the disk, it is

necessary to determine the acceleration, and the rotational inertia, of the track by itself so this

rotational inertia can be subtracted from the total, leaving only the rotational inertia of the disk.

1. To do this, take the disk off the rotational apparatus and repeat Finding the Acceleration of Disk

and Track for the track alone.

➧ NOTE: It will take less “friction mass” to overcome the new kinetic friction and it is only necessary

to put about 30 g over the pulley in Finding the Acceleration of Disk and Track.

Disk Using Ball Bearings (Free Disk)

Mount the disk upside-down at the same radius as before. Now the ball bearings at the center of the

disk will allow the disk to rotate relative to the track. Repeat Accounting For Friction and Finding

the Acceleration of Disk and Track for this case and record the data in Table 4.2.

Calculations

Record the results of the following calculations in Table 4.3.



2. Calculate the experimental value of the rotational inertia of the fixed disk and track combined. 3. Calculate the experimental value of the rotational inertia of the track alone.

4. Subtract the rotational inertia of the track from the rotational inertia of the fixed disk and track. This

will be the rotational inertia of the fixed Table 4.3: Results disk alone.

5. Calculate the experimental rotational inertia for fixed disk -and track combined 6. Subtract the rotational inertia of the track from the rotational inertia of the free disk and track. This

will be the rotational inertia of the free disk alone.

Rotational Inertia for Track Alone

Rotational Inertia for Fixed Disk

Off-Axis (experimental value)


and Track Combined

Rotational Inertia for Free Disk Alone (experimental value


Off-Axis (theoretical value)

Rotational Inertia for Point Mass

(theoretical value)

% Difference for Fixed

% Difference for Free Disk

7. Calculate the theoretical

value of the rotational

inertia of the fixed disk off

axis. 8. Calculate the theoretical

value of a point mass

having the mass of the

disk.

9. Use a percent difference to

compare the experimental

values to the theoretical

values.

Table 4.3: Results

24

EQUIPMENT REQUIRED

-Smart Pulley Timer Program - balance

-Rotational Inertia Accessory (ME-8953)

-Rotating Platform (ME-8951)

-Smart Pulley Photogate

Purpose

A non-rotating ring is dropped onto a rotating disk and the final angular speed of the system is

compared with the value predicted using conservation of angular momentum.

Theory

When the ring is dropped onto the rotating disk, there is no net torque on the system since

the torque on the ring is equal and opposite to the torque on the disk. Therefore, there is no

change in angular momentum. Angular momentum is conserved.

L =Iiω i =If ω f

where Ii is the initial rotational inertia and ω

i is the initial angular speed. The initial rotational

inertia is that of a disk 2

12

1RMI i

and the final rotational inertia of the combined disk and ring is

)(2

1

2

1 2

2

2

12

2

1 rrMRMI f

So the final rotational speed is given by

ifrrMRM

RM

)( 2

2

2

12

2

1

2

1

Setup

1. Level the apparatus using the square mass on the track.

2. Assemble the Rotational Inertia Accessory as shown in

Figure 5.1. The side of the disk with the indentation for the

ring should be up. 3. Mount the Photogate on the metal rod on the base and

position it so it straddles the holes in the pulley on the

center rotating shaft. Smart Pulley

4. Start the DataStudio program. Select „Smart Pulley

(Rotational) as the sensor.

5. Set up a Graph display of Velocity (rad/s) versus Time (s)

Rotational Disk

(indentation up)

"A" base

Experiment 5: Conservation of Angular Moment

rotational disk

(indentation up)

Figure 5.1: Assembly for

Dropping Ring onto Disk

25

Procedure

1. Hold the ring just above the center of the disk. Give the disk a spin using your hand

2. Start recording data. After about 25 data points have been taken, drop the ring onto the spinning

disk See Figure 5.2.

photogate head

"A" base

Figure 7.2: Experiment Setup

3. Continue to take data after the collision for a few seconds and then stop recording data.

4. Examine the Graph display of the rotational speed various time. Use the Autoscale tool to

resize the axes if necessary. 5. In the Graph display, use the Smart Tool to determine the angular velocity immediately before

and immediately after the collision. Record these values in Table 5.1. 6. Weigh the disk and ring and measure the radii. Record these values in Table 5.1

Analysis Table 5.1: Data and Results

1. Calculate the expected

(theoretical) value for

the final angular

velocity and record this

value in Table 5.1

2. Calculate the percent

difference between the

experimental and the

theoretical values of the

final angular velocity


Initial Angular Speed

Final Angular Speed (experimental value)

Mass of Disk

Mass of Ring

Inner Radius of Ring

Outer Radius of Ring

Radius of Disk

Final Angular Speed (theoretical value)

% Difference Between Final Angular Speeds

Figure 5.2: Experiment Setup

dropped ring

rotational disk

26

Questions

%KE Lost =2

22

2

12

1

2

1

ii

ffii

I

II

1. Does the experimental result for the angular speed agree with the theory?

2. What percentage of the rotational kinetic energy lost during the collision? Calculate this and

record the results in Table 5.1.

27

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